Least upper bound, or union

Consider the graphs

represented by a digon, and

represented by a triangle. The loops of the first have length 2, those of the second, length 3. It would seem that loops belonging to one or the other would belong to their union, in which links would exist if and only if they had already existed in the components.

The sense of inclusion in the definition of the order relation was chosen so that one of the two connected components could be ignored while making comparisons. In general, it is profitable to consider a graph as a union of its disconnected parts and to observe that conversely, nontrivial unions are disconnected.

The topological matrix of the union of two diagrams is the direct sum of the topological matrices of the constituents.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx